Applications of set theory to abstract harmonic analysis

集合论在抽象调和分析中的应用

• 批准号: RGPIN-2017-05712
• 负责人: Steprans, Juris
• 金额: $ 2.19万
• 依托单位: York University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759135
• 关键词: Applications; set; theory; abstract; harmonic

The reasons for the remarkable effectiveness and applicability of mathematics have been the subject of various philosophical considerations, but the question becomes even more difficult when the use of extra mathematical axioms come into play. How could an axiom whose truth is not easily verifiable have any influence on practical mathematical matters such as those one encounters in physics or finance? But anyone who thinks there can be no such influence should consider an axiom that is often used without further consideration in mathematics, the axiom of infinity. All that the axiom of infinity says is that the set of all natural numbers exists; not that each natural number exists on its own, but that the set of all of them exists as a mathematical object.One should not be too quick to dismiss this as a philosophical issue with no practical applications. Every time we use an irrational number, such as the square root of 2, we are implicitly using the axiom of infinity since we are manipulating all of the infinitely many digits of that number at once, as a single set. Infinity cannot be avoided, even in geometry; the completeness of the real number line requires it. Most mathematicians have no difficulty accepting the axiom of infinity as a valid axiom, although it must be admitted that there are some who feel that applications of mathematics that invoke infinity cannot be trusted. But what is to be said about axioms beyond the axiom of infinity, axiom that do not usually appear in mathematical arguments? While it is rare, there are some ramifications of such axioms in applied areas such as quantum physics and even economics and financial mathematics. Can such arguments be trusted? Are these axioms needed? What can we prove (and, hence, be confident in the truth of) in mathematics without assuming these axioms?The purpose of the proposed research program is to help delineate those areas of mathematics that are susceptible to axioms beyond the commonly accepted ones. Determining this boundary is an important first step in understanding which mathematical arguments might be problematic because of their reliance on extra axioms.

数学有效性和适用性的原因一直是各种哲学考虑的主题，但是当使用额外的数学公理开始发挥作用时，这个问题变得更加困难。一个公理如何不容易验证的公理会对实际数学问题（例如物理或金融遇到的人）有任何影响？但是，任何认为没有这种影响的人都应该考虑一个经常使用的公理，而无需进一步考虑数学，即无穷大的公理。无限的公理表明，所有自然数的集合都存在。并不是说每个自然数量都存在，而是所有自然数量都作为数学对象而存在。一个人不应该太快将其视为没有实际应用的哲学问题。每当我们使用一个非理性数字（例如2的平方根）时，我们都隐含地使用无穷大的公理，因为我们是一次单一的一组操作，一次无限地操纵该数字的所有数字。即使在几何形状中，也无法避免无限。实际数字行的完整性需要它。大多数数学家毫无困难地接受无穷大的公理作为有效的公理，尽管必须承认有些人认为数学的应用是无法信任的。但是，关于无限公理之外的公理，通常不会出现在数学论点中的公理呢？虽然很少见，但在量子物理，甚至经济学和金融数学等应用领域中，这种公理会产生一些影响。这样的论点可以信任吗？这些公理需要吗？在数学中，我们可以证明（因此，对）在不假定这些公理的情况下（拟议的研究计划的目的）是什么？确定这个边界是理解哪种数学参数可能会出现问题的重要第一步，因为它们依赖于额外的公理。